Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(14,16)(17,18,21,22)(19,20,23,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,40)(34,35)(38,39);(11,15)(14,16)(17,19,21,23)(18,20,22,24)(26,29)(28,37)(31,34)(33,38);(2,3)(6,7)(9,10,12,13)(11,16,15,14)(17,19)(18,20)(21,23)(22,24)(25,27,30,35)(26,31,29,34)(28,38,37,33)(32,40,36,39);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,32)(27,33)(29,36)(30,37)(31,39)(34,40)(35,38);(1,4)(2,6)(3,7)(5,8)(25,29,30,26)(27,34,35,31)(28,36,37,32)(33,40,38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926694);